The first time it happened, we were in Grade 11: six or seven teenagers, in line at a doughnut shop. I felt an urgent tug at my sleeve.
“Cynthia,” my friend D. whispered, pulling me away. “Do I have enough for coffee?”
He opened his palm to reveal a damp cluster of nickels, dimes and quarters. It was far more than the 50 cents needed for a cup, but D. had no way of knowing that.
Now, I wasn’t much of a math student myself, but at least I could count change. Later, I found out that several others in the group had also been acting as after-school accountants for D. – an eloquent, funny boy who nonetheless lacked even the most basic arithmetic skills.
How many people like D. walk among us? There may be many. According to a recent federal government report, only half of Canadians have the numeric skills and knowledge “necessary to function well” in society. Illiteracy is a much discussed problem, but its sibling – innumeracy – goes relatively unnoticed. Perhaps it’s because we have long internalized the idea that, while reading is necessary, mathematical ability is the gift of a chosen few.
“Stand firm in your refusal to remain conscious during algebra,” Fran Lebowitz once wrote. “In real life, I assure you, there is no such thing.” Lebowitz was wrong, of course: mathematical reasoning is necessary all through life, affecting decisions we make in personal finance, travel, cooking and real estate, to name just a few. Our collective inability to analyze data has left us at the mercy of politicians and their advisers, who likely also quail in the face of math. “Innumeracy,” wrote math professor Lynn Arthur Steen in his 1997 book, Why Numbers Count, “perpetuates warfare, harms health and weakens families.”
Today, the Canadian public school system still turns out teenagers who are left puzzled at the cash register. Earlier this year, the Toronto Star reported that one-third of community college students in Ontario are in danger of failing first-year math. This could be linked to the fact that one-third of all high school students are currently registered in applied (non-academic) courses. A 2004 study showed that 60 per cent of these students were also failing, or close to it.
A small cohort of kids have always “just gotten it,” but the situation is frightening for those who don’t – especially in a work climate where some firms now administer standardized math tests to prospective employees.
Newfangled instructional methods are always being tried in an effort to right things. The 1960s saw the introduction of “new math” as a panicked response to Russia’s Sputnik program, which was thought to be the result of the country’s superior educational system. The method confused parents, was roundly mocked by satirists such as Tom Lehrer and Charles Schulz, and died within 10 years. More recently, President George W. Bush convened the National Mathematics Advisory Panel to arrest the lag in American science and engineering capability. (In 2006, the United States scored 25th out of 30 countries on the math portion of the Program of International Student Assessment. Canada, reassuringly, scored 7th.) Its final report advocated yet another overhaul of how math is taught.
John Mighton (BA 1978 VIC, MSc 1994, PhD 2000) would agree that the state of North American math education is far from ideal. The eminent playwright, teacher, author and U of T adjunct math professor has long been the Chicken Little of arithmetic, using words such as “disaster” to describe the state of student numeracy in Canada’s public schools. The tutoring program he started in his Toronto apartment 10 years ago (known as JUMP, short for Junior Undiscovered Math Prodigies) was originally designed for struggling inner-city kids, but is now gaining respect in classrooms as far afield as England and South Africa.
Mighton is routinely portrayed as a back-to-basics saviour, rescuing children from the inquiry-based (also known as reform) math that is currently the fashion in Toronto schools. It’s a reputation that makes JUMP attractive to those of a more conservative educational bent. But while it’s true that his system stresses pencil-and-paper work, learning concepts in steps, computational fluency and extensive review, it would be wrong to say he wants things the way they used to be.
More than anything, Mighton says, we need to change the culture of self-defeat that persists in so many math classrooms. The change he seeks is as psychological as it is pedagogical. “As early as Grade 3 the kids notice which kids in the class are smarter, and which ones are less capable academically,” says the 50-year-old polymath, who’s also the author of two books (The Myth of Ability and The End of Ignorance) that reinforce this point. Mighton contends that the time-honoured practice of sorting children into A, B and C groups only worsens math phobia: “By the time kids are streamed in high school, they’ve lost any motivation to do math or engage in it.” Two separate studies on JUMP, conducted by the Ontario Institute for Studies in Education at U of T, suggest that the program increases math confidence.
Despite this, Mighton can’t get a break at home. This past May, consultants employed by the Toronto District School Board reached into their pencil cases and jabbed him with a set square. The board issued a position paper on his methods, condemning JUMP as a form of “rote, procedural learning where students memorize very small computational steps with little to no conceptual understanding.” This is another way of saying that Mighton wants to give students the topdown directive that 4 x 4 = 16, instead of guiding them to discover on their own that 16 is equal to four groups of four.
Documents like this infuriate Mighton (to the extent that he can be infuriated: quiet and firm, he shares a birthday with Gandhi). The statement, he says, is “completely full of misinformation. They’ll say this is the correct way to teach – but we didn’t say that wasn’t the correct way to teach! We actually advocate the same thing, but they’ve never looked closely at our program.”
The Toronto board adheres to the reform style of teaching that has become increasingly popular over the last 15 years. Reformers tend not to be as pencil-and-paper-bound as Mighton is. They believe a dependence on traditional algorithms prevents children from understanding the processes behind a given problem. To this end, they often link math concepts to real-life situations, use concrete objects and encourage group work. They emphasize the importance of understanding mathematical concepts over computational ability, stressing the “why” over the “how.”
But are the two camps really so far apart? Thumbing through Mighton’s Grade 4 workbook, I see that he, too, advocates the use of materials such as card games, coins and fingers. He also stresses that a concept be learned before an algorithm (such as long division) is introduced. “John Mighton is a friend of mine,” says reform math advocate Barry Onslow, a retired professor of education who contributed to the development of the current Ontario math curriculum: in short, someone you’d think would be Mighton’s enemy. “People want to have this polarization,” he says. “They want things to be black or white. They aren’t. There are many shades of grey.”
Nevertheless, other reformers still view Mighton as Mr. Drill and Kill. Some Toronto teachers who want to use JUMP are specifically told not to by their principals, and have to sneak its tenets into their lessons. This is not the case in Vancouver, where JUMP methods are openly embraced. “Are people more backward out west?” Mighton asks. “Why is it we are being welcomed there, when people are forced underground here? And would they welcome our program if they weren’t getting results?”
A similar battle is being waged in California, the centre of what’s become known as the Math Wars. For almost two decades now, reform and basics advocates have been at each others’ throats, amassing research and mounting publicity campaigns to support their respective sides. Remember the Reading Wars, which pitted whole-language modernists against phonics traditionalists? Trade letters for numbers and you get the idea.
Some reformers condemned the National Mathematics Advisory Panel’s final report, released this spring. They thought it was too rigidly basics-oriented in nature. Reading it, however, one sees that its language is quite even-handed: “To prepare students for algebra, the curriculum must simultaneously develop conceptual understanding, computational fluency and problem-solving skills,” it says. “Debates regarding the relative importance of these aspects of mathematical knowledge are misguided. These capabilities are mutually supportive.”
The fact is that traditionalists such as Mighton and reformers such as Onslow share many ideas. Perhaps the way to mathematical salvation lies in seeing what they have in common, instead of what divides them. We forget that when a war is being waged, children know about it. In the 1970s, students such as D. and myself saw “new math” disparaged every day in cartoons and newspapers. It made us mistrust the entire subject.
If there is one idea that unites the math fixers, it is this: Children must understand what they are doing. Today, D. would likely be diagnosed with a learning disability, but no matter. Both Mighton and Onslow believe that if he’d only been taught why he was performing mathematical operations, he too could have succeeded. “Why do we invert and multiply?” Mighton asks a group of teachers gathered for one of his training sessions on a sunny spring morning. We are all learning how to divide fractions, something we’d done in Grade 7 and now have completely forgotten. There is a rule for it, but no one’s ever taken it apart to show us the machinery. Mighton does, and we all breathe a sigh of relief before putting the idea into practice ourselves. We do not move on until everyone gets it.
Barry Onslow’s approach is more direct, but that might not be a reform thing – that might be Barry Onslow. “What’s 10 divided by a half?” he asks me. “If I said that to you, what might be your instant reaction?”
“My instant reaction would be to freeze up completely,” I tell him.
“But if I said you have to give me an answer…”
“Oh, all right then,” I stammer. “Five.”
“That is the answer that most people would give,” he says, “because you think of 10 sheared in half. But that’s not what it means. What it means is, how many halves are there in 10 wholes?”
Even I can easily see that it’s 20; divorced of meaningless symbolism, the idea makes perfect sense. But will I remember this later? “Your experience in school was one of memorizing rules,” Onslow tells me. “And if you forget those rules, then you forget your math. But if mathematics makes sense, you don’t need a lot of rules, because it fits into a framework.”
The scary reality remains, however, that math failure seems to persist for some kids, no matter what instructional method is used. When I speak to my own children’s math teachers, they complain about everything except methods. Scarce resources, disparate knowledge among students and their own anxiety are much bigger problems.
Children in Asian countries persistently score higher on international tests than their counterparts in North America, but this is not necessarily because they are taught differently. Maybe they just study harder. With more homework, shorter holidays and fewer extracurricular distractions, it’s said that an average Chinese pupil enters university with two more years of education than a Canadian one.
And teacher quality is important. “I read this paper recently where they looked at factors such as gender and socioeconomic background, and they found that everything was washed out by teacher effect,” Mighton says. “If you have a good teacher, all those other things aren’t so important.”
Regardless of the teaching method, learning math is like walking a tightrope – it requires sharp attention and a constantly active mind. One slip and you fall off. Expecting each student in a class of 30 to follow along at the same pace is asking a lot, especially in an era of video games and three minute music videos.
“The research in psychology suggests that attention is the key to everything,” Mighton says. “If you’re not harnessing attention, keeping it focused, then the brain is not developing. I think a lot of things that used to train attention, like how to stay on task, to focus, how to look for patterns in things – all those deeply important cognitive skills – aren’t being fostered in kids. One of the effects is that kids have more attention problems now.”
Culturally, we have also done children a disservice by depicting mathematicians as geniuses, lunatics or both. Movies such as A Beautiful Mind and Good Will Hunting (in which John Mighton played a small part, both as a consultant and an actor) and the television show Numb3rs all command public interest, but their heroes tend to be tortured loners who sit above and apart from the rest of us.
According to Onslow, math is anything but a solitary pursuit. “How do we learn?” he asks. “Do we learn in isolation? We have to get our ideas out in the open where people can discuss them. That way, they can see why some of their thinking might be faulty.” Similary, Mighton stresses active, rather than passive learning, though JUMP is less language-based than most reform classes (so that, among other things, ESL students will not be at a disadvantage).
It should be mentioned that Mighton himself was no child math prodigy. “Sometimes I did well, but sometimes I did terribly,” he says. Although he eventually earned a PhD in mathematics, he almost failed calculus in his undergraduate years at the University of Toronto. None of this surprised him. As a child, a glance into his older sister’s psychology textbook had convinced him that geniuses reveal themselves early; he was not one.
Mighton turned to the consolations of philosophy, earning a master’s degree in the subject from McMaster University in 1978. It was during this time that he developed his own life philosophy: that genius – or at the very least, extreme success – might just be something you can program. He read Sylvia Plath’s Letters Home, which was revelatory. “It was clear that she taught herself to write by sheer determination,” he says. “She did everything she could to train herself, including writing imitations of the poems she liked.”
Plath’s method worked for Mighton, and he has since become one of the country’s most successful playwrights. His approach is strikingly methodical. For Mighton, a so-called creative work does not erupt like Krakatoa in the artist’s mind. It is a tightly constructed puzzle, given shape and nurtured by the proofs of those who’ve gone before. “There are patterns, structure and logic in the arts,” he says. “It’s very similar to mathematics.”
His is a highly democratic vision of genius, open to the hardworking and the strongly encouraged as well as the gifted. Ironically, this is something Mighton figured out on his own. The traditionalist’s journey, then, has actually been reformist in nature, built on ideas he discovered and mastered for himself. It’s an example of how patterns and the connections between them can be found in so much of what we do – proof that mathematical thinking, rather than the memorization of formulas, is an essential skill to develop.
Every good math teacher agrees on at least one thing: math is a subject of lifelong importance, and not a childhood torture on the order of bra-snapping or dodgeball. Onslow advocates family math nights to reacquaint math-anxious adults with the subject. Mighton believes in later math learning, particularly in a climate of rising concern about brain fitness in the aging population. “It’s almost an evangelical experience when adults go back and find that they’re smart enough, because many people carry a great burden of feeling stupid from school. It has a great psychological effect. People forget the adjustments they went through to deal with this stuff,” Mighton says.
“Don’t know much about algebra,” Sam Cooke once sang; “don’t know what a slide rule is for.” Cooke thought the mere knowledge that one and one is two would be all he needed for true happiness, but math educators such as John Mighton and Barry Onslow think differently. They believe that happiness and advanced math are not necessarily mutually exclusive. One day, the rest of the world may even agree with them. Now, what a wonderful world that would be.
Cynthia Macdonald is a writer in Toronto.
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3 Responses to “ Fear of Numbers ”
Cynthia Macdonald notes that she found the language of the National Mathematics Advisory Panel’s final report “quite even-handed,” though some reformers condemned it for being too “rigidly basics-oriented.”
Good math instructors have always strived to simultaneously develop conceptual understanding, computational fluency and problem-solving skills. The fact that these so-called reformers equate their misconceptions about how mathematics is taught and learned with conceptual understanding has always been the heart of the problem.
Professor of Mathematics
California State University, Los Angeles
When my husband, Ron Sawatsky (PhD 1986), and I moved to the U.S. in 2000, I went back to college and had to take a couple of math courses. I was petrified because although I am good at arithmetic, I have a mental block when it comes to conceptual problems. One course, called “The History and Culture of Math,” helped me realize that my fear of math was based on the way I learn; I need a context to understand a problem. In the course, we learned why math is useful and how it developed historically, and worked on math problems as they related to historical events. After finishing the course, I still didn’t love math. But I had learned to appreciate the subject in a different way – and to fear it less. If, in elementary or high school, I had been introduced to math via history and culture, my mental block might never have developed.
Susan M. Sawatsky